\section{ Skew Codes }

Recently F. Ulmer and co-authors \cite{paper:SQcodes},\cite{paper:SQcoding} have focused attention on the construction of error-correcting codes over (non-commutative) skew polynomial rings. Their work is summarized in this section including preliminary and background information required in order to understand skew codes.

\subsection{ Skew Polynomial Rings }

Let $F$ denote the Galois Field $GF(q)$ of order $q$, an extension of a field $K$ with characteristics $p$, where $q=p^{r}$ for some $r \in \mathbb{N}$. Let $\theta \in Auto(F)$ be an automorphism defined over $F$. The skew polynomial ring $F[x;\theta]$ is the ring of polynomials defined as:
\[
F[x;\theta]=\{a_{0}+a_{1}x+ \dots + a_{n}x^{n}|a_{0},a_{1}, \dots ,a_{n} \in F \}
\]
where addition is component-wise but multiplication follows the following property:
\[
xa=\theta(a)x
\]

It can be readily seen that this ring is {\bf not} in fact commutative as over $GF(4)=\{0,1,\alpha,\alpha^{2}\}$ for instance we have:
\[
(x^{2}+\alpha x + \alpha^{2})(x+1) \neq (x+1)(x^{2}+\alpha x + \alpha^{2})
\]

The ring $F[x;\theta]$ is a left and right euclidean ring whose left and right ideals are principal \cite{paper:Ore}. Here right division means that for $P_{1},P_{2} \in F[x;\theta]$ which are non zero, there exist unique polynomials $Q_{r},R_{r} \in F[x;\theta]$ such that:
\[
P_{1} = Q_{r}.P_{2} + R_{r}
\]
If $R_{r} = 0$ then $P_{2}$ is a right divisor of $P_{1}$ in $F[x;\theta]$. The definition of left divisor is similar using the left euclidean division. In the ring $F[x;\theta]$ left and right gcd and lcm exist and can be computed using the left and right euclidean algorithm.

\begin{Thrm}
An element $P \in F[x;\theta]$ is {\bf central} (i.e. commutes with all elements of $F[x;\theta]$) if and only if $P= \sum{i=0}{m} c_{i}x^{i.\alpha}$ where $\alpha = |<\theta>|$.
\end{Thrm}

The central elements of a skew ring are the generators of two sided ideals which are used to generate codes.

\subsection{ Codes As Ideals in Skew Rings }

The following propositions stated without proof from \cite{paper:SQcodes} and \cite{paper:SQcoding} generalize the above concepts to generate codes from skew rings.

\begin{Lemma}
Let $F_{q}$ be a finite field of a prime power order $q$, $\theta$ an automorphism of $F_{q}$ and $n$ an integer divisible by the order $|<\theta>|$ of $\theta$. Then the ring $F_{q}[x;\theta]/(x^{n}-1)$ is a principal left ideal ring in which left ideals are generated by $G$, a right divisor of $x^{n}-1$ in $F_{q}[x;\theta]$.
\end{Lemma}

\begin{Thrm}
Let $F_{q}$ be a finite field of a prime power order $q$, $\theta$ an automorphism of $F_{q}$, and $\mathcal{C}$ a linear code over $F_{q}$ of length $n$. If $|<\theta>|$, the order of $\theta$, divides $n$, then the code $\mathcal{C}$ is a $\theta$-cyclic code if and only if the skew polynomial representation of $\mathcal{C}(x)$ of $\mathcal{C}$ is a left ideal $(G) \subset F_{q}[x;\theta]/(x^{n}-1)$. 
\end{Thrm}

The skew polynomial rings, unlike the commutative case, are not Unique Factorization rings. So in comparison, there are more right divisors $G$ in skew rings, making the class of skew codes far more larger than commutative rings. This rich class of codes has resulted in the discovery of many codes having parameters that improve upon those of previously best-known codes.  




